Finite Group Databasebeta

← #35306905

Subgroups — #35306905

Order 21248.

474 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918808C211yesmaximal, characteristic
2#24917268C212yesmaximal
3#24905919C212yesmaximal
4#24905338C211yesmaximal, characteristic
5C83×#1984C211yesmaximal, characteristic
6#19128258C2212yes—
7#19128008—24no—
8#19127997—22no—
9#19127996—22no—
10#19127986—22no—
11#19127942—22no—
12#19127912—22no—
13#19127741—22no—
14#19127725C2212yes—
15#19127710—22no—
16#19127433—24no—
17C2×C4×Q8×C83C2211yescharacteristic
18C2×C4⋊C8×C83C2211yescharacteristic
19C2×C4×C8×C83C2211yescharacteristic
20#336×C83—22no—
21C2×Dic332—22no—
22C2×#13050574—22no—
23#14581792D412yes—
24#14581792D412yes—
25#14581784—22no—
26#14581771—22no—
27#14581771—22no—
28C4×Dic166—22no—
29C4×Dic166—22no—
30#14581751—22no—
31#14581726—22no—
32#14581725—24no—
33#14581723—24no—
34C4⋊(C83⋊C8)—22no—
35C4⋊(C83⋊C8)D414yes—
36#14581717—24no—
37#14581716—24no—
38#14581716—24no—
39C22×Q8×C83D411yescharacteristic
40C22×C8×C83—21no—
41C22×C8×C83D412yes—
42C4×Q8×C83—22no—
43C4×Q8×C83—22no—
44C2×C4⋊C4×C83—21no—
45C2×C4⋊C4×C83D411yescharacteristic
46C2×C42×C83—21no—
47C2×C42×C83C2311yescharacteristic
48C4⋊C8×C83—22no—
49C4×C8×C83—22no—
50C2×Dic166—22no—
51C2×Dic166—22no—
52Dic332—82no—
53#13050574—42no—
54#13050574—42no—
55C2×C83⋊C8—24no—
56C4⋊Dic83—22no—
57C4⋊Dic83—22no—
58C4⋊Dic83—22no—
59C4⋊Dic83—22no—
60Dic83⋊C4—42no—
61Dic83⋊C4—42no—
62C22⋊Dic83—42no—
63C4×Dic83—42no—
64C2×Q8×C83—21no—
65C2×Q8×C83—22no—
66C2×Q8×C83—22no—
67C2×Q8×C83Q1612yescomplemented
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83C2×D411yescharacteristic
71C22×C4×C83C2×D411yescharacteristic
72C22×C4×C83#4111yescharacteristic
73C2×C8×C83—22no—
74C2×C8×C83—22no—
75C2×C8×C83—41no—
76C2×C8×C83—41no—
77C2×C8×C83—42no—
78C4⋊C4×C83—22no—
79C4⋊C4×C83—42no—
80C42×C83—41no—
81C42×C83—41no—
82C42×C83C2×D412yes—
83C42×C83C2×D412yes—
84C23×C83#12111yescharacteristic
85Q8×C83—24no—
86Q8×C83—42no—
87C2×C4×C83—21no—
88C2×C4×C83—21no—
89C2×C4×C83—22no—
90C2×C4×C83—41no—
91C2×C4×C83—41no—
92C2×C4×C83—41no—
93C2×C4×C83—42no—
94C2×C4×C83—41no—
95C2×C4×C83#12011yescharacteristic
96C2×C4×C83#12012yes—
97C2×C4×C83#12112yes—
98C2×C4×C83#12711yescharacteristic
99C2×C4×C83C2×Q1611yescharacteristic
100C2×C4×C83#13511yescharacteristic
101C2×C4×C83#13712yes—
102C2×Dic83—42no—
103C2×Dic83—42no—
104Dic166—44no—
105Dic166—42no—
106Dic166—42no—
107C8×C83—42no—
108C8×C83—42no—
109C83⋊C8—44no—
110C22×C83—21no—
111C22×C83—21no—
112C22×C83#39211yescharacteristic
113C22×C83#46711yescharacteristic
114C22×C83#47011yescharacteristic
115C4×C83—21no—
116C4×C83—22no—
117C4×C83—42no—
118C4×C83—81no—
119C4×C83#45112yes—
120C4×C83#45212yes—
121C4×C83#45511yescharacteristic
122C4×C83#45611yescharacteristic
123Dic83—82no—
124Dic83—82no—
125#10259—831nomaximal, Sylow
126C2×C83—21no—
127C2×C83—21no—
128C2×C83#200111yescharacteristic
129C2×C83#200711yescharacteristic
130C2×C83#201711yescharacteristic
131#3249—831no—
132#3078—832no—
133#1984D8311yescharacteristic
134#1493—832no—
135#1292—831no—
136C83#1025911yescharacteristic, complemented, Sylow
137#533—832no—
138C2×C4×Q8C2×D8311yescharacteristic
139#479—1664no—
140#477—1662no—
141#477—1662no—
142#475—1662no—
143#467—1662no—
144#462—1662no—
145#428—1662no—
146#423—832no—
147C2×C4⋊C8C2×D8311yescharacteristic
148#419—1662no—
149C2×C4×C8C2×D8311yescharacteristic
150#336—22no—
151Q8⋊C8—1664no—
152C22×Q8D33211yescharacteristic
153C2×Q16—1662no—
154C2×M16—1662no—
155C22×C8—21no—
156C22×C8C22⋊D8312yes—
157C4⋊Q8—832no—
158C4⋊Q8—832no—
159#125—1662no—
160C22⋊Q8—1662no—
161C22⋊Q8—1662no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—1662no—
165C4×Q8—1662no—
166#117—1662no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4D33211yescharacteristic
169C2×C42—21no—
170C2×C42C22×D8311yescharacteristic
171#107—1664no—
172#107—1662no—
173#106—1664no—
174C4⋊C8—22no—
175C4⋊C8—834no—
176C4⋊C8—1662no—
177#103—1664no—
178#103—1664no—
179#103—1664no—
180C4×C8—22no—
181C2×Q8—21no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—1662no—
185C2×Q8—1662no—
186C2×Q8Dic33212yescomplemented
187C22×C4—21no—
188C22×C4—21no—
189C22×C4C2×C22⋊D8311yescharacteristic
190C22×C4C2×D33211yescharacteristic
191C22×C4#1305059011yescharacteristic
192Q16—6642no—
193M16—3322no—
194M16—3322no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—42no—
200C2×C8—1664no—
201C4⋊C4—22no—
202C4⋊C4—42no—
203C4⋊C4—3322no—
204C4⋊C4—3322no—
205C4⋊C4—1662no—
206C4⋊C4—1662no—
207C4⋊C4—1662no—
208C4⋊C4—1662no—
209C22⋊C4—3322no—
210C42—41no—
211C42—41no—
212C42—3322no—
213C42D4×D8312yes—
214C42D4×D8312yes—
215C23#1458176611yescharacteristic
216Q8—24no—
217Q8—42no—
218Q8—3324no—
219Q8—3322no—
220Q8—3322no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—3322no—
230C2×C4—3322no—
231C2×C4#1458176011yescharacteristic
232C2×C4#1458176212yes—
233C2×C4#1458176512yes—
234C2×C4#1458178811yescharacteristic
235C2×C4C2×Dic33211yescharacteristic
236C2×C4#1458181411yescharacteristic
237C2×C4#1458182512yes—
238C8—42no—
239C8—42no—
240C8—3324no—
241C22—21no—
242C22—21no—
243C22#1912761011yescharacteristic
244C22#1912794211yescharacteristic
245C22#1912795911yescharacteristic
246C4—22no—
247C4—21no—
248C4—42no—
249C4—81no—
250C4—6642no—
251C4—6642no—
252C4#1912785912yes—
253C4#1912786512yes—
254C4#1912787811yescharacteristic
255C4#1912788311yescharacteristic
256C2—21no—
257C2—21no—
258C2#2490847311yescharacteristic
259C2#2490850911yescharacteristic
260C2#2490856711yescharacteristic
subgroupG/H
centerC22#19127610
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1984C2
socleC22×C83#392
prime radicalC23×C83#121
hypercenter#1984D83
hypocenterC83#10259
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1984

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
67. C2×Q8×C83Q16semidirect
136. C83#10259semidirect
186. C2×Q8Dic332semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#249188082
#249172682
#249059192
#249053382
C83×#19842
#1025983